We know this via sacred tradition, and it's proven by the historical timelines of all churches today who claim to be "Christian". We know that the early Christians weren't Lutherans, for example, because Luther didn't start his church until 1517-ish. That's an obvious one, but in the same manner, trace all 30,000+ protestant religions backwards throughout time and they all disappear by the time you hit 1500, which is about 1500 years to late to be qualified as Christ's Church. The process of elimintation leaves only one Christian church: the CC. Jesus founded one church and one church only. There are other ways to establish the CC as Christ's church, but tracing history alone is sufficient (and simplest ).
As far as the first-century Christians being Catholic, we know this through sacred tradition (of which the NT is a form of). First of all, the word "catholic" is derived from the greek for "universal". Salvation history in the OT was not universal, but built up to it. The first covenant between God and man was between a couple (Adam & Eve). Then it grew to a family (Noah). Then to a tribe (Moses). Then to a nation (Abraham), but was not yet universal. Even Isaiah longed for the Messiah to establish a covenant for all nations (Isaiah 2:2-4) Since Jesus then founded one church for all mankind, by its nature it is universal, and so a proper name for it is "Catholic Church". The early church fathers testify to the beauty and necessity of the early church, which was most definitely Catholic:
Interestingly, St. Augustine (as quoted above) was Pope at the council of Hippo in 393 AD which presented the 27 books of the NT as canonical (to the exclusion of hundreds of other scriptures circulating at the time).Ignatius of Antioch
"Let no one do anything of concern to the Church without the bishop. Let that be considered a valid Eucharist which is celebrated by the bishop or by one whom he ordains [i.e., a presbyter]. Wherever the bishop appears, let the people be there; just as wherever Jesus Christ is, there is the Catholic Church" (Letter to the Smyrneans 8:2 [A.D. 110]).
The Martyrdom of Polycarp
"And of the elect, he was one indeed, the wonderful martyr Polycarp, who in our days was an apostolic and prophetic teacher, bishop of the Catholic Church in Smyrna. For every word which came forth from his mouth was fulfilled and will be fulfilled" (Martyrdom of Polycarp 16:2 [A.D. 155]).
The Muratorian Canon
"Besides these [letters of Paul] there is one to Philemon, and one to Titus, and two to Timothy, in affection and love, but nevertheless regarded as holy in the Catholic Church, in the ordering of churchly discipline. There is also one [letter] to the Laodiceans and another to the Alexandrians, forged under the name of Paul, in regard to the heresy of Marcion, and there are several others which cannot be received by the Church, for it is not suitable that gall be mixed with honey. The epistle of Jude, indeed, and the two ascribed to John are received by the Catholic Church (Muratorian fragment [A.D. 177]).
Tertullian
"Where was [the heretic] Marcion, that shipmaster of Pontus, the zealous student of Stoicism? Where was Valentinus, the disciple of Platonism? For it is evident that those men lived not so long agoin the reign of Antonius for the most partand that they at first were believers in the doctrine of the Catholic Church, in the church of Rome under the episcopate of the blessed Eleutherius, until on account of their ever restless curiosity, with which they even infected the brethren, they were more than once expelled" (Demurrer Against the Heretics 30 [A.D. 200]).
Augustine
"We must hold to the Christian religion and to communication in her Church, which is Catholic and which is called Catholic not only by her own members but even by all her enemies. For when heretics or the adherents of schisms talk about her, not among themselves but with strangers, willy-nilly they call her nothing else but Catholic. For they will not be understood unless they distinguish her by this name which the whole world employs in her regard" (The True Religion 7:12 [A.D. 390]).
"We believe in the holy Church, that is, the Catholic Church; for heretics and schismatics call their own congregations churches. But heretics violate the faith itself by a false opinion about God; schismatics, however, withdraw from fraternal love by hostile separations, although they believe the same things we do. Consequently, neither heretics nor schismatics belong to the Catholic Church; not heretics, because the Church loves God, and not schismatics, because the Church loves neighbor" (Faith and Creed 10:21 [A.D. 393]). (quotes taken from http://www.catholic.com/library/What_Catholic_Means.asp)